The numerical solution of the second Painlevé equation.

*(English)* Zbl 1172.65037
Summary: The Painlevé equations were discovered by Painlevé, Gambier and their colleagues during studying a nonlinear seccond-order ordinary differential equation. The six equations which bear Painlevé’s name are irreducible in the sense that their general solutions cannot be expressed in terms of known functions. Painlevé has derived these equations on the sole requirement that their solutions should be free from movable singularities. Many situations in mathematical physics reduce ultimately to Painlevé equations: applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics, and fiber optics. This fact has caused a significant interest to the study of these equations in recent years. In this study, the solution of the second Painlevé equation is investigated by means of Adomian decomposition method, homotopy perturbation method, and Legendre tau method. Then a numerical evaluation and comparison with the results obtained by the method of continuous analytic continuation are included.

##### MSC:

65L05 | Initial value problems for ODE (numerical methods) |

34M55 | Painlevé and other special equations; classification, hierarchies |

35Q53 | KdV-like (Korteweg-de Vries) equations |

37K10 | Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies |